Single-Observation Uniformity Testing under Increasing Precision via Lacunary Harmonics

TL;DR

This paper introduces a novel uniformity test for a single high-precision observation using lacunary harmonic analysis, enabling detection of multiscale structures invisible to traditional digit-frequency methods.

Contribution

It develops a multiscale harmonic digit expansion framework and a quadratic test statistic with a chi-square null distribution for single-observation uniformity testing.

Findings

The test statistic converges to a chi-square distribution under the null hypothesis.

The method detects multiscale harmonic deviations from uniformity.

Detectability improves with increasing observation precision.

Abstract

A test of uniformity on [0,1] is developed for the setting of a single observation recorded with sufficient precision. Although consistency against general alternatives is not attainable with only one draw in the classical large-sample sense, a multiscale harmonic digit expansion provides a framework for structured inference. By aggregating trigonometric components across digit scales at Hadamard-gap frequencies, a quadratic test statistic is constructed whose null distribution converges to a chi-square law via a lacunary central limit theorem. Under departures from uniformity, the statistic is driven by Fourier components induced by digit-scale transformations of the observation, with detectability depending on their coherent accumulation as precision increases. The resulting procedure detects multiscale harmonic structure that remains invisible to classical digit-frequency methods.